How many ways are there to arrange N spheres in space, so they form a rigid cluster?

A cluster is rigid if it cannot be continuously deformed by any small but finite amount, and maintain all pairs in contact. Equivalently, it is an isolated solution to the system of equations listing the pairs of spheres that are exactly touching, after fixing the translational and rotational degrees of freedom. This is a nonlinear notion that is a natural set of clusters to consider, because these clusters are metastable when the spheres interact with a short-range potential.

Unfortunately there is no efficient way to test for rigidity computationally. However, there are methods to test for stronger versions. All clusters listed below are pre-stress stable (to numerical tolerance.) This is slightly stronger than second-order rigidity, but not as restrictive as infinitesimal rigidity (a linear notion).


The following data was generated by an algorithm that follows all possible one-dimensional transition paths between rigid clusters.

The files (once unzipped) are txt files, with one cluster listed per line.

Please let me know if you find new clusters! (or, gasp, mistakes.) I will update this page as new data is generated.

Last updated August 22, 2014.

All coordinates

Ground states (clusters with the maximum number of contacts)

All clusters for n<10 are ground states.

Most clusters were sharpened afterward using Bertini, so the coordinates of all but the hypostatic clusters (those with fewer than 3n-6 contacts) and those for N=14,19 are accurate to floating-point precision. Bertini can be used to sharpen these to arbitrary precision, so if you would like higher precision please let me know. The hypostatic and singular clusters should be accurate to roughly 3e-8, and the regular clusters for N=14,19 to roughly 9e-16.

For n=15—19, only clusters with a certain minimum number of contacts are listed. The goal here was to find the ground states: those with the maximum number of contacts.